I am very confused about what $\nabla$ signifies when used to describe affine connections.

In his book Riemannian Geometry, Manfredo do Carmo defines an affine connection as follows. Let $\mathcal{X}(M)$ denote the set of all vector fields of class $C^{\infty}$ on $M$. An affine connection $\nabla$ on differential manifold $M$ is a mapping $\nabla : \mathcal{X}(M) \times \mathcal{X}(M) \rightarrow \mathcal{X}(M)$ which is denoted by $(X,Y) \xrightarrow{\nabla} \nabla_{X}Y$ and which satisfies the following properties....

...and he goes on to state what an affine connection is. But I have also seen the covariant derivative written as $\nabla_{X} Y$ where $Y$ is being differentiated in the direction of $X$.

My question: Is $\nabla$ being used to represent two different things (ie (1) affine connection as some sort of map and (2) the covariant derivative)? Or perhaps, am I not understanding the relationship between the covariant derivative and affine connections? What's going on here?


EDIT: I'm just learning these concepts. A novice. But I will venture a guess and say I mean the following definition of the covariant derivative

The covariant derivative of a type $(r,s)$ tensor field along $e_c$ is given by the expression:

$ (\nabla_c T)^{a_1 \ldots a_r}{}_{b_1 \ldots b_s} = \frac{\partial}{\partial x^c}T^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}+\,\Gamma ^{a_1}{}_{dc} T ^{d \ldots a_r}{}_{b_1 \ldots b_s} + \cdots + \Gamma ^{a_r}{}_{dc} T ^{a_1 \ldots a_{r-1}d}{}_{b_1 \ldots b_s} -\,\Gamma ^d {}_{b_1 c} T ^{a_1 \ldots a_r}{}_{d \ldots b_s} - \cdots - \Gamma ^d {}_{b_s c} T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s-1} d}$

  • $\begingroup$ It may be helpful to include the definition of covariant derivative that you are using. $\endgroup$ Dec 29 '14 at 22:45
  • $\begingroup$ Not sure I'm knowledgeable enough, but I gave the definition I have seen in the books I've read. $\endgroup$ Dec 29 '14 at 23:06

There is a connection between the covariant derivative and affine connections: the latter generalize the former.

In particular, the covariant derivative you described depends on a Riemannian metric, while you can consider affine connections of manifolds without a metric.

You have surely noticed that the conditions imposed on affine connections are usual properties of the covariant derivative.


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